Short summary of recent events in Libya:
* Rebel and Gaddafi forces face east/west stalemate
* Gaddafi's forces organize and rally, pushing Rebels to Benghazi
* At some point, SAS (and probably CIA) inserted into Libya
* UN approves no-fly zone and "all necessary measures" to protect civilians
* Gaddafi declares ceasefire, declares willingness to talk to opposition, meanwhile moving forces into Benghazi
* Anti-Rebel operations continue in Benghazi (in efforts to "last-minute" end the Rebel forces)
* French strikes begin on land targets (tanks, etc)
* US/UK strikes begin on anti-air, fuel, airport, communication/command sites
* Arab League condemns ground attacks
"All necessary measures" is likely a deliberately vague clause, allowing the West to justify (at least to itself) nearly any military action, while abiding by its post-Iraq rule-following sensibilities.
Of course, the protection of civilians is a political mask over the true intent: regime change in Libya. As much as the Rebels are a bit of an unknown factor (and some of them are clearly Islamist), the West is quite tired of Gaddafi.
The implications are numerous and highly variable.
West-Middle East Relations
The West certainly hopes that its pro-Rebel action will endear it to the new anti-establishment movements and regimes that seem to be sweeping the Middle East. The West gets to claim "we helped without occupying!" when the new governments of the Middle East start making serious decisions. The fact that the Rebels asked consistently for Western help is a good sign that the PR has the potential to be good.
That said, the Arab league is condemning the ground strikes, saying that the West has already gone past its mandate of protecting civilians. Indeed, the "protecting civilians" focus of the UN resolution may be a political crutch to real action. The Arab League, moving towards a popular/ democratic rhetoric, is worried that the air strikes will lead to non-insignificant civilian casualties; these governments thus do not want to be perceived as supporting more Western action in the Middle East that seems to damage Arab/Muslim people.
The two questions as to the PR outcome will be the # of civilian casualties and the outcome of the campaign. More casualties will be tolerated in the short-term if Gaddafi's brutal regime is removed and replaced by something more friendly, but only if the front-line action is driven by the Rebels themselves. The West will need to keep some of its guns in check in order to avoid looking less like humanitarians and more like imperialists.
Part of the major risk is scope creep, though it depends on the country.
The United States has already committed to take a back seat after the Suppression of Enemy Air Defenses (SEAD) campaign ends, allowing countries like France and Italy to take over enforcing the no-fly zone and preventing ground forces from entering Benghazi en masse.
That said, the risk is creating a stalemate, in which a withdrawal of air forces will allow Gaddafi to succeed, but an endless no-fly zone campaign might quickly frustrate both domestic European opinion and Middle East opinion.
To break that stalemate and seriously cripple Gaddafi's forces (thus giving the rebels the advantage) will require action beyond the original mandate, though this can often be done with enough discretion that only minor objections will be thrown up.
The West's goal here may be to make sure a friendly country, probably Egypt, is committed to supporting the rebels and making sure they can be rearmed, fed, and regrouped during the air campaign, such that they can make a sustained push west towards Tripoli when the initial SEAD campaign ends; Western forces are likely to be able to provide key intelligence to help the Rebels find isolated pro-Gaddafi units and overtake them quickly and decisively, which would further arm the rebels with artillery and armor.
Likely military outcomes here are either a stalemate or a rebel victory. Gaddafi will struggle to get the forces he needs into Benghazi (and struggle to control them well enough to effectively occupy key population centers).
As a default, military units are now hiding in heavily populated areas, though if they're otherwise paralyzed, it may give Rebel forces there an opportunity to harass the units into a state of unpreparedness for a more sustained Rebel counter-attack.
While a Rebel victory is possible, it's also unlikely. The Rebels lack military leadership, and (unlike Yemen or Egypt) have not had major elements of the armed forces defect and turn their guns against the leadership. Until that happens, the Rebels would have to completely knock out the sizable Libyan army in order to claim victory over Tripoli.
If the stalemate persists, the Rebels are likely to keep their alternative government in the east, and a de facto split of Libya would occur. The Benghazi government could garner international support; especially from Egypt (which would want influence over the large oil reserves in that area).
but whether the Rebels win or not, the big question of its political makeup remains a major concern for the West. Clearly, the Rebels are anti-despotic, but what else? Libya had a brutal campaign against extremist Islam for some time--how strong are these influences in the Rebel group? Do the Rebels desire a secular state? Will it be Western-friendly? These are all questions left unanswered, and will largely determine the amount of sustained Western support for the government.
Western Cohesion and Trans-Atlantic Relations
Perhaps most interesting is that the biggest benefit the West is likely to get is its own internal sense of dignity and goodwill. With US-led operations in the last 10 years going militarily poorly and morally questionably, there have been a lot of bad feelings and aimlessness among the West, even with respect to itself. The West's strategic direction is confused and frustrated (especially as Russia seems to be running circles around NATO's internal conflicts). If Libya goes well, the West can achieve a "feel-good" victory like it had in the early 1990s, and a renewed confidence in its ability to be an effective and relatively-welcomed police force.
Western military cohesion will not be that of the early 1990s, but it may improve. Specifically, the US will have an opportunity to act in an effective support role, both improving relations with Europe (by allowing Europe to lead the operation that's most relevant to its immediate periphery), and setting up a system in which the US can limit its engagement. The US has far and away the best logistical/transport capacity and control, as well as the best battlefield intelligence. These two factors make it a perfect candidate to support other Western nations' campaigns against less impressive forces, and the US can benefit significantly by setting up alliances in which it maintains that role.